Covering with Hecke points
type: Item
instance of: scholarly article
title: Covering with Hecke points
zbMATH Open document ID: 0839.11022
author: Patrick Chiu
published in: Journal of Number Theory
publication date: 1995-10-22T00:00:00Z
review text: Let \({\mathcal H}^n\) be the \(GL_n (\mathbb{R})\) upper-half-space \({\mathcal H}^n = GL_n (\mathbb{R})/O(n, \mathbb{R})Z\), where \(O(n, \mathbb{R})\) is the orthogonal group and \(Z\) the subgroup of scalar matrices. Consider the quotient space \({\mathcal M}_n = SL(n, \mathbb{Z}) \backslash {\mathcal H}^n\). One may endow this space with a natural metric. The author studies the distribution of Hecke points in \({\mathcal M}_3\). For \(p\) a prime, the \(p\)-Hecke points are by definition the cosets represented by the upper triangular integral matrices of determinant \(p\). Focusing on the case \(n = 3\), he shows that given a bounded set \(\Omega\) of \({\mathcal M}_3\) there exists a number \(N_\Omega\) such that for all primes \(p > N_\Omega\) any point of \(\Omega\) is inside a ball of radius \(Kp^{- 1/10}\) centered at some \(p\)-Hecke point. Here \(K\) is an absolute constant. In particular, this implies that the Hecke points are dense. The exponent \(- 1/10\) is based on the estimate for the Hecke eigenvalues of the \(GL_3\) cusp forms due to \textit{H. Jacquet}, \textit{I. I. Piatetski-Shapiro}, and \textit{J. Shalika} [Ann. Math., II. Ser. 109, 169-212 (1979; Zbl 0401.10037)]; the Ramanujan conjecture would give exponent \(- 1/5\). The author remarks that the same proof works when \(n = 2\), and, using the bound of \textit{D. Bump}, \textit{W. Duke}, \textit{J. Hoffstein} and \textit{H. Iwaniec} [Int. Math. Res. Not. 1992, No. 4, 75-81 (1992; Zbl 0760.11017)], gives a similar result with radius \(Kp^{-9/56}\). The proof is based on harmonic analysis on \({\mathcal M}_3\). For \(\varepsilon > 0\), the author shows that there is a point-pair invariant \(K_\varepsilon (x,y)\), corresponding to a positive definite integral operator, which is nonzero only when the distance \(d(x,y) < \varepsilon\). He then applies the Hecke operator \(T_p\) to this function (in \(y)\), and analyzes the result by means of the \(GL_3\) Spectral Theorem to obtain his theorem. T…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 792467
zbMATH Keywords: approximation of matrices
zbMATH Keywords: point-pair invariant
zbMATH Keywords: \(p\)-Hecke points
zbMATH Keywords: distribution of Hecke points
zbMATH Keywords: harmonic analysis
zbMATH Keywords: positive definite integral operator
MaRDI profile type: Publication
full work available at URL: jnth.1995.1077
OpenAlex ID: W2050228135
DOI: 10.1006/JNTH.1995.1077
description: scientific article; zbMATH DE number 792467